Area of a Rectangle given Perimeter?

Answer 1

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A farmer has 3600 feet of Fencing which he wants to use to enclose a rectangular corral and then divide it into 10 smaller, equally-sized corrals with fences parallel to the sides (See Below). Find the largest area that can be enclosed.

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I know that 3x+6y=3600 and that x*y=a but can't go beyond there...HELP!!!

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Referring to your drawing, the greatest area is enclosed by making the corral

[ 1/6 Fence ] = 600-ft wide by [ 1/12 Fence ] = 300-ft high.

Horizontal fence = 3 runs @ 600-ft = 1,800-ft

Vertical fence = 6 runs @ 300-ft = 1,800-ft

Total fence = 3,600 ft

Total area = (600 x 300) = 180,000 sq ft = 4.132 acres (rounded)

Each pen = 120-ft wide x 150-ft high = 18,000 sq ft = 0.4132 acre (rounded)

I got this in the normal way of using calculus ... setting equal to zero the derivative

of area (in terms of length of fence) with respect to one dimension ... but I don't

have any other way to prove that it's the greatest area. I did notice something

interesting, though. This doesn't prove anything, but it feels like maybe I'm holding

a secret in my hand that I can't read:

-- We know that for a fixed perimeter, the greatest possible area with straight sides

is a square. If the perimeter is 3600-ft ... (all the fence you have) ... then the square

would have sides of 900-ft, and the area would be (900 x 900) = 810,000 sq ft,

without sub-dividing.

Now stay with me here:

-- Instead of two horizontal runs of fence for a single area, we're using three ... 1.5 times as much.

-- Instead of two vertical runs of fence for a single area, we're using six ... 3 times as much.

-- (1.5 x 3) = 4.5 .

-- We could have had 810,000 sq ft without sub-dividing. Divide that by 4.5

and you get 180,000 ... just what we wound up with after sub-dividing.

Does this mean anything ? I don't know. But take your 180,000 sq ft and

use it in good health. That's my answer and I'm sticking to it.